Difference between revisions of "Positive-definite kernel"
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| − | + | A complex-valued function $ K $ | |
| + | on $ X \times X $, | ||
| + | where $ X $ | ||
| + | is any set, which satisfies the condition | ||
| − | The theory of positive-definite kernels extends the theory of positive-definite functions (cf. [[Positive-definite function|Positive-definite function]]) on groups: For a function | + | $$ |
| + | \sum _ {i,j= 1 } ^ { n } K( x _ {i} , x _ {j} ) | ||
| + | \lambda _ {i} \overline \lambda \; _ {j} \geq 0, | ||
| + | $$ | ||
| + | |||
| + | for any $ n \in \mathbf N $, | ||
| + | $ \lambda _ {i} \in \mathbf C $, | ||
| + | $ x _ {i} \in X $ | ||
| + | $ ( i = 1 \dots n) $. | ||
| + | The measurable positive-definite kernels on a measure space $ ( X, \mu ) $ | ||
| + | correspond to the positive integral operators (cf. [[Integral operator|Integral operator]]) on $ L _ {2} ( X, \mu ) $; | ||
| + | in order to include arbitrary positive operators in this correspondence one has to introduce generalized positive-definite kernels, which are associated with Hilbert spaces [[#References|[1]]]. | ||
| + | |||
| + | The theory of positive-definite kernels extends the theory of positive-definite functions (cf. [[Positive-definite function|Positive-definite function]]) on groups: For a function $ f $ | ||
| + | on a group $ G $ | ||
| + | to be positive definite it is necessary and sufficient that the function $ K( x, y) = f( xy ^ {-} 1 ) $ | ||
| + | on $ G \times G $ | ||
| + | is a positive-definite kernel. In particular, certain results from the theory of positive-definite functions can be extended to positive-definite kernels. For example, Bochner's theorem is that each positive-definite function is the Fourier transform of a positive bounded measure (i.e. an integral linear combination of characters), and this is generalized as follows: Each (generalized) positive-definite kernel has an integral representation by means of so-called elementary positive-definite kernels with respect to a given differential expression [[#References|[1]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> M.G. Krein, "Hermitian positive kernels on homogeneous spaces I" ''Ukr. Mat. Zh.'' , '''1''' : 4 (1949) pp. 64–98 (In Russian)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> M.G. Krein, "Hermitian positive kernels on homogeneous spaces II" ''Ukr. Mat. Zh.'' , '''2''' : 1 (1950) pp. 10–59 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> M.G. Krein, "Hermitian positive kernels on homogeneous spaces I" ''Ukr. Mat. Zh.'' , '''1''' : 4 (1949) pp. 64–98 (In Russian)</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> M.G. Krein, "Hermitian positive kernels on homogeneous spaces II" ''Ukr. Mat. Zh.'' , '''2''' : 1 (1950) pp. 10–59 (In Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)</TD></TR></table> | ||
Latest revision as of 08:07, 6 June 2020
A complex-valued function $ K $
on $ X \times X $,
where $ X $
is any set, which satisfies the condition
$$ \sum _ {i,j= 1 } ^ { n } K( x _ {i} , x _ {j} ) \lambda _ {i} \overline \lambda \; _ {j} \geq 0, $$
for any $ n \in \mathbf N $, $ \lambda _ {i} \in \mathbf C $, $ x _ {i} \in X $ $ ( i = 1 \dots n) $. The measurable positive-definite kernels on a measure space $ ( X, \mu ) $ correspond to the positive integral operators (cf. Integral operator) on $ L _ {2} ( X, \mu ) $; in order to include arbitrary positive operators in this correspondence one has to introduce generalized positive-definite kernels, which are associated with Hilbert spaces [1].
The theory of positive-definite kernels extends the theory of positive-definite functions (cf. Positive-definite function) on groups: For a function $ f $ on a group $ G $ to be positive definite it is necessary and sufficient that the function $ K( x, y) = f( xy ^ {-} 1 ) $ on $ G \times G $ is a positive-definite kernel. In particular, certain results from the theory of positive-definite functions can be extended to positive-definite kernels. For example, Bochner's theorem is that each positive-definite function is the Fourier transform of a positive bounded measure (i.e. an integral linear combination of characters), and this is generalized as follows: Each (generalized) positive-definite kernel has an integral representation by means of so-called elementary positive-definite kernels with respect to a given differential expression [1].
References
| [1] | Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian) |
| [2a] | M.G. Krein, "Hermitian positive kernels on homogeneous spaces I" Ukr. Mat. Zh. , 1 : 4 (1949) pp. 64–98 (In Russian) |
| [2b] | M.G. Krein, "Hermitian positive kernels on homogeneous spaces II" Ukr. Mat. Zh. , 2 : 1 (1950) pp. 10–59 (In Russian) |
Comments
References
| [a1] | H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968) |
How to Cite This Entry:
Positive-definite kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_kernel&oldid=48250
Positive-definite kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_kernel&oldid=48250
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article